Solution of some problems of neutron transport theory by the method of generalized eigenfunctions

SOLU’TION OF SOME PROBLEMS OF NEUTRON TRANSPORT THEORY BY THE METHOD OF GENERALIZED V. P. GORELOV,

EIGENFUNCTIONS*

V. I. IL’IN and V. 1. YUFEREV Moscow

5 April 1971; revised CASE’S

version

7 September

1971)

method is used as the basis for solving the neutron transport equation in

plane and spherical geometries. The solution may be used to find the critical parameters of a system of fissionable materials.

Introduction It is of practical interest to develop analytic methods for solving the neutron transport equation in the context of an energy-dependent collision kernel and anisotropic

scattering.

In the present paper the solution is sought as a series in the complete transport system of eigenfunctions of the transport equation; for single-velocity theory, these functions were described by Case in [l]. Case’s

method was used in [2] to solve the transport equation in a plane

geometry, in the context of an energy-dependent collision kernel and without any simplifying assumptions regarding the form of this kernel. The results obtained were so complicated, difficult.

however, that practical

application

of them is

The first step towards simplifying the solution is to make the collision kernel Sdegenetate” with respect to energy and the cosine of the scattering angle; this should lead to complete or partial reduction of the problem to an algebraic form. We shall not discuss degree of degeneracy *Zh.

vychisl.

the question of convergence

of the collision

Mat. mat. Fiz.,

kernel;

12, 5. 1245-1264.

with respect to the n-th

instead, we shall extend the 1972.

Some problems

technique solving the transport equation by Case’s energy-dependent

collision

problem “degenerate”

kernel and anisotropic

with respect to suitable

In Section 1 we find the critical assuming an energy-dependent isotropic scattering.

201

of neutron tmnsport theory

method to the problem with scattering

by making the

variables.

dimension of a plate of fissionable

collision

material,

kernel and using the approximation of

Xn Section 2, Part 1 we find the neutron spectrum in a plate when it is irradiated from outside by a given neutron flow. We assume anisotropic scattering and an energy-dependent collision kernel. In Section 2, Part 2 we extend the resuits of 133 to the case of a multilayer sphere, when, in each individual layer, we have to allow for anistropic scattering

up to an expansion term linear in the cosine of the scattering

angle.

A common feature of the application of Case’s method to the problems mentioned is as follows: substitution of the expression for the neutron flow in the boundary conditions and the form of the eigenfunctions of the continuous part of the spectrum of eigenvalues enable a singular integral equation to be obtained in the spectral function of this part of the spectrum, while the condition for solvability

of this equation enables us to obtain the coefficients

expansion into discrete sion of the system.

eigenfunctions,

Notice finally that the system of eigenfunctions complete in the single-velocity is exactly

of the flow

and when required, the critical

approximation;

dimen-

was shown in cl] to be

since the proof of completeness

the same for our problems, it will be omitted.

1. Determination of the critical dimension of a plate in the approximation of isotropic collisions The problem of finding the critical

dimension of a plate of fissionable amounts to the following:

material in the approximation of isotropic collisions to find the solution of the transport equation

V. P. Gorelou,

202

which is positive

at all points

the outer surfaces

Since

V. I. ll’in and V. 1. Yuferev

of the medium considered,

the equations

(1 .l) and (1.2)

2a has to be found from solvability The notation cosine

in (1 .l) and (1.2)

of the angle

direction section section

between

of the x axis, of interaction

will be assumed

is:

below.

I,!J(x, p, E) is the neutron flow, p is the of neutron

the neutrons

motion and the positive

Z is the effective

and the nuclei

to be energy-independent,

drift from the unit energy in the neighbourhood

interval

total cross-

of the medium, which &(E’

-f E) is the cross-

in the neighbourho~

of E’

of E, and +a are the coordinates

of

boundaries.

It should

be mentioned

that, since

are 0 and m, the kinematics

nature of the nuclear introducing obtaining

on

the plate dimension

which will be stated

E is the neutron energy, between

into the unit interval

interval

are homogeneous,

conditions,

the direction

for simplicity

of neutron

the plate

and which satisfies,

of the plate, .the condition

reactions

suitable

single

the conjugate

The solution

the energy

of elastic

collision

have to be reflected

powers,

limits

of the collision and the threshold

in the kernel

which must be taken

X:,(E’

into account

-f E) by when

equation.

of (1.1)

will be sought

as a superposition

of plane waves:

T, (x, p, EJ = e=x’t Q, (6 c”, E).

(1.3) After substituting

(1.3)

in (1 .l),

we get

(1 - cl/‘t)ct, (t, /J, E) = 0.5H 0, E),

(1.4) where

(1.5)

fj(t,E)=

jdp. 1B”(E;-E) a,(t, p’,E’)dE’. -1

As in single-velocity classes,

according

(1) the class L-1, 11:

0

problems,

the solutions

to the range of variation of functions

of (1.4)

split

of the parameter

bounded in the interval

into two

t:

-1 < p < 1, with t = z f

Same problems

of neutron transport

G(z, p, E)=

(1.6)

(2) the class

of functions

singular

zH (2, El -

2k --id

203

theory

f

in the interval

-1 < ~1< 1, with t E

(-1, 1):

a 0,

(1.7)

p, El =

tH 0, E)

Here, H (t, E) and h (t, E) are functions class),

and also,

(1.2) with respect

bounded

are even in t, by virtue to the substitution

The functions

(1.6) relate

+ x (t, E)6 (t - p>.

20 “-PI in the given interval

of the symmetry

of equations

(the H(1.1) and

(x, p) + (-x, -p).

to the discrete

part of the t spectrum,

since,

when substituted in (l.S), a homogeneous integral equation in H (z, E) is obtained, which is well known to be solvable with only certain values of the parameter

2:

Meantime, equation

if (1.7) is substituted

is obtained

in (1 S), an inhomogeneous

(1.9)

N(t,E’)dE’=

=

s0

h(t, E’)

It might be possible solve

integral

in H (t, E):

the last equation

equation equation

z, (E’ * E) z dE’,

t E (-1,l).

to find h(t, E!) from physical for H (t, E).

considerations,

is best used only at an intermediate connecting H (t, E) and X (t9 E).

stage ,of the problem,

It may happen in general that the homogeneous integral equation to (1.9) has a solution for certain values of the parameter t belonging (see t21):

then

But it will be shown below that this latter as an

conjugate to (-1, 1)

204

V. P. Gotelov, Hence,

by Fredholm’s

V. 1. Il’in

Theorem

h (t, E) here makes its appearance,

for those

values

and V. 1. Yuferev

4 (see e.g. [4]), an auxiliary

condition

on

namely,

of the parameter

t at which the solution

This

of (1 JO) exists.

situation, i.e. the introduction of discrete eigenvalues into the interval (-1, I), can be allowed for in the way described in [2]. It will therefore be assumed for simplicity

in future that (1.10)

The spectrum

of Equation

method of “degenerate” When the kernei

is never solvable

(1.8) can be found approximately

kernels

is represented

in eigenvalues

det IG-

the

in this way, it is best to use Laguerre

z z

the definition

of these

t-1, 11 can be written

2

(1.11)

by using

[4].

polynomials L, (E) as basis functions; given in fS] will be used below. The equation

for t E. (-1, 1).

f

zfn[(z+l)/(z-_)I

)

polynomials

in this case

as

=o,

where

IlGll ’ Ri-k!(

)J 200

oL,t (E) e-To+ (E) dE,

0s (E) = fL< (A”) x0 (‘L*

E, dE’,

0

and 1 is the unit matrix, polynomial consider

k, i = 0, 1,. . . ,

in the expansion the n-th approximation

n,

while n is the order of the leading

of &,(E’ --fE) into Laguerre polynomials, of the method of degenerate kernels.

i.e. we

Since the left-hand side of (1.11) is an even function in z, the roots of the equation will appear in pairs rt L,, p = 1, 2,. . . , M. If we fix z = L, in (1 .S), If (L,, E) will be in general defined apart from a constant, which may be combined with the corresponding coefficient of the expansion of li, (x, p, E) in T,(x,

CL,IQ. In the n-th approximation

of the method of degenerate

kernels,

therefore,

Some problems

the space-angle

distribution

of neutron transport

205

theory

$ (x, p”, E) can be written

as

(1.12)

where we have taken account hp = b_,, and the integral principal value [61. Notice

of the problem

symmetry,

is to be understood

that functions

by virtue

in the sense

(1.6) and (1.7) form a system

of which

of the Cauchy

of non-orthogonal

functions. The set of functions l@+ (t, 11, E)] orthogonal to this system may be obtained by replacing in the transport equation (1.1) a/& by -a/&, &(E’ by &(E + E’ ), and ‘I’&, p, E)” E) as an expansion in

Tit (x, When constructing can prove useful:

/L,

+ E)

by $’ (x, p, E)’ , after which we seek $’ (x, ~1,

E) = e-x’t @+ (x, /.L,E).

Green’s

function,

orthogonality

the following

property

sdEs

d~@+(t’,~,QdD(t, p,E)=&r,

0

which means

--i

with each Q(t, I_‘,E) is associated

that,

one-to-one

a function

a’+ (4 cc, EL In our case,

of the n-th approximation

of the method of degenerate

kernels,

the proof that the system of eigenfunctions @ (+L,, ~1,E), @(t, p, E) is complete is just the same as in 111 for the single-velocity approximation, or in [21 for any type of collision kernel the proof will be omitted.

in a plane

To sum up, in the framework degenerate

kernels,

orthogonal

functions.

singular

this expression

integral

with energy

of the n-th approximation

(1.12) is an expansion

After substituting following

geometry

equation

in a complete

in the boundary

dependence;

of the method of system

conditions

(SIE) can be obtained

hence

of non-

(1.2), the

in H (t, E):

V. P. Gorelov,

206

V. 1. Il’in

and V. I. Yufetev

B (t, E’) dE’ =

-_-

“2, (E’ -+ E) c

J

dE,

0

MbP[TLph Ii4lm+

E p=i

+TLp (-a,

,p,,m++ j t(3

x

-E’)

_ e-2”/‘)

“Z,(E”

J 0

o

dt X

t+Ipl

>:

where Bft,

E) =

H(t,

E)e”““; [. . .]dE’.

In just the same way as in single-velocity be reduced to a Riemann boundary value function on the cut (-1, l), the solvability as orthogonality [-1, 11, relative

where H’ (z,,

with z = zP E

131, this last SIE can

problem on the jump of an analytic conditions for which may be written

conditions for the eigenfunctions to the Cauchy integral of #‘(It\,

E) satisfy,

problems

of the operator E):

A, * (z), z g

c-1, 11, the equation

A,’ (z) H+ (z, E) = H+ (z, E) -

and clearly, This

H+ (5 E) = wt (_G E). last equation

so that we have

has to be solved

in the same n-th approximation

as (1.8),

Some problems

on neutron transport

2

C

G’--

i

zln[(z+1)/(2-_)I

I

207

theory

c+(z)=o,

where

ah(E) t+Li (E) dE;

K+ Wll~ = c,*tz 1 are

the coefficients

H+(z,E)=

in the expansion

$C,l(z)L,(E)e-‘. k=O

It can easily

be seen

In addition,

that the eigenvalues

since

F(jtl,

E) is an even function

zp = ,I,, is identical

with (1.13)

need be considered,

e.g.

precisely

single-velocity

problems

Omitting

the half with zp = L,;

working,

to the solution

of t, we find that (1.13) only half of these

with

equations

the number of equations

left is

b,.

we quote the final expression of the integral

for H (t), where H (t) is the vector of W (t, E) in a series

equation

1-r +]A-

(t) ]-I}

A (2) -t

obtained

with bounded

formed from the coefficients

in tik (E):

= O.;Je-““” ([h+(t)

-I- (25~3) -‘e-““” where

i.e.

[3].

the intermediate

ki(t)

are the same

for H (t, E) can be found by the same method as is used in

as first approximation kernel

with zp = -L,,

equal to the number of unknown constants

The equation

expansion

of the last equation

so that we have zp = +I,,P, p = 1, 2, . . . , M, in (1.13).

as the roots of (l.ll),

of the

V. P. GoreEou, V. I. Win and V. 1. Yufereu

208

IIA(t)ll, = A,

(t) and A, (t) are the coefficients

&$&)o,(E)=

-j

k-0

ZO(E;-+)

of the expansion )(

0

X {~wLp~--a,

Itl,E’)+T-L+z,

Itj,E’)]}dE’,

p-1

which are linearly dependent on the b . Notice that the factor in front of the integral in the expression for H (t) is finite at t = 0, since ht (0) = h’(O) = 1. Substitution

of the first approximation for H (t, E) in the system of equations

(1.13) leads to a system of homogeneous equations in the bp, the condition for solvability

of which in fact determines the critical

After finding the $

plate dimension.

and hence H (t, E), we find X 01, E) from the boundary

condition (1.2): X (p, E) -_ -e-aM

cbP1~Lp(--a,!l~ltE)+~-Lp(--a,

1&E)]--

p=*

By this means we are able to find all the quantities defining the positive stationary neutron distribution in a plate of fissionable material, the critical dimension of which is found during the course of solution,

from the solvability

condition for the relevant Riemann boundary value problem. Notice that, as in the single-group and multi-group problems on finding the critical dimension, it is to be expected that at least one pair of imagina~ roots + iL will appear among the roots of the characteristic equation (1 .ll) for the fissionable “standing

material; these imaginary roots ensure the appearance of partial * waves in the space distribution of the neutrons.

Meantime the integral term in the expansion (1.12) is to be treated as an interference term, describing the influence of the boundaries, since, as a increases, both H (t, E) and X (t, E) decrease in absolute value. Furthermore, it may easily be shown that, in the presence of external radiation, the function h (t, E) will contain a term ~(LL,E)e*‘\” 1 (where f (/.L,E) is the neutron flow from vacuum on to the outer surfaces of the plate), describing the distribution of the neutrons of the external radiation, which fail to be scattered through the plate.

on passing

of neutron transport theory

Some pro&le~s

if the kernel system

&(E’

+ E) of the collision

of eigenfunctions

{a’@,

integral

or, E)j transforms

209

is symmetric,

into the direct

the conjugate system

i@(t, CL,

E)j; this somewhat facilitates solution of the problem, since it then becomes unnecessary to find H’ (z, E). This happens e.g. when considering a plate of non-fissionable material, irradiated by a flux of thermal neutrons, when account must be taken

of the thermal

pressure

of the nuclei

of the material.

It is known

(see e.g. [7]) that, when account is taken of the thermal pressure of the material nuclei, the form of the kernel in the collision integral is such that a transformed kernel can be introduced, which proves to be energy-symmetric. It should be mentioned that, if the formal approach outlined in the present section is used, the technique of solution in 131 for single-velocity problems in laminated

plane

problems

and spherical

involving

We shall

geometries

an energy-dependent

next turn to problems

must be taken

can easily collision

be extended

to similar

integral.

in which anisotropy

of the neutron

scattering

into account.

2. Solution of the neutron transport equation in the context of anisotropic scattering 1. DETERMINATION OF THE NEUTRON ENERGY SPECTRUM IN A PLATE, USING A TWICE DEGENERATE COLLISION KERNEL We shall the neutron

confine spectrum

ourselves

in Part 1 of this section

in a plate

irradiated

will be used for the collision Our problem

0

xj

amounts

externally.

to the problem

A twice degenerate

of finding kernel

kernel.

to solving

the integrodifferential

equation

J(n)

c

g;“’(zq h’d”’(II’) j

F’, (P’)$ (z, P’,E’) 4~’ a’.

The function I,$(x, p, E), positive within the plate, has to satisfy on its faces the following boundary condition, implying that the system in question sub-critical:

is

210

V. P. Gorelov,

(2.2)

qoqGzO,E)

The notation

=f(CLSO,E).

in (2.1)

2, (E’ * E)

and (2.2)

denotes scattering

angle,

of simplicity, notation

Y

the n-th energy

harmonic

free path length As before,

(

in the expansion

of the kernel radiation,

to be an even function

as in Section

Xs

Pn (x) in the cosine

polynomials

E) is the flux of external

will be assumed

is as before;

gin)(E) h:) (E’)

k=i

in Legendre and f@,

is as follows:

J(n)

= y.

a.5

Cl)/ c into a series

V. I. ll’in and V. I. Yuferev

(E’ , !A’ + E,

x: = !JQ’

which,

of the

for the sake

of ~1. The remaining

1, we use the approximation

whereby

the

is energy-independent. I+!J(x, /I, E) will be sought as a superposition

of plane waves

(1.3). After substituting the eigenfunctions

(1.3)

the following

expressions

the properties

Z-CL cn=o

are obtained

mentioned

for

in Section

J(n)

N (2n+l)

O(z,y,E)=Z

(2.3)

in (2.1),

Q (t, p, E), possessing

p7l (cl> c

2

s??) (E)HP

(Z))

P.=,

ZE [-l,l]; (2.4) k=.,

where,

for the entire

I&@)(t)=

Substitution

(2.5)

range of variation

Sdpi~p,(pr)hl;n) 0 --I

of (2.3)

of t, we define

(E’)O(t,

in the last equation

leads

p’,E’)dE’.

to an equation

h (z)H (2) = [I - zc (z)lH (2) = 0, z g

which is only solvable

when .z satisfies

the equation

L-1, 11,

for H$n) (2):

1:

Some probEems

211

of neutron transport theory

det [I - ZC (2X! = 0.

(2.6) In these

last relationships, K(z)

/In =

N,(z),

n =

1, 2,. ..,

(N$- l),

i@,, (2) 4 = HP (z), k = 1,2,. . . , i(a), IIC(2) IInn’ =Cnn~(Z),

n, n’=l,

2I,..,

(N+l),

I [2(n’

- 1)3-

1]Q,_,(Z)P&,(Z)j g:f-“(E)h:“-”(E)dE, ”

n 2 d,

IIC,,~(2) iI& =i ~

I

12 (n’ - 1) + l]P,_, (2) Qn’__l(z) J gy-1’ (E) Q-1) (8) dE, 0 n-cd;

and QneI (z) is a Legendte function of the second are connected by the linear transformation

tip)

(2.7) which, 1).

in view of the symmetry

In view of this, The left-hand

eigenvalues

appear

On substituting connecting

@.8)

(2) = (-1rH;“’ of the external

kind.

Notice

C-z), radiation,

also holds in t E

the X(t, E) in (2.4) is an even function side of (2.6) proves in pairs

+L,,

that the Hfitn’ (2)

(-I,

of t.

to be an even function

of 2, so that its

p = 1, 2,. . . , M.

(2.4) in the definition

of Hi”’

(t), the following

equation

H (t) and X (t, E) is obtained:

i,(t)o(t)=[r_tc,(t)Irr(t)=jI(1,E)h(t,E)dI3, 6

where jjh (t, E)ll,

= h, (t, El, Ilh,(tt E)llk = P,_,@)h$“-*j(E),

and the matrix C, is the same as C except

that the ln[(z + l)i’(z

-

I)] in the

212

V. P. Gorelov,

expression

V. I. Il’in md

V. I. Yufereu

for Qk (z) is replaced by In [(l + t)/ (1 -

For each z satisfying

t)l.

the equation (2.6), we can find N (z) up to a constant,

and, as in Section 1, this constant can be combined with the corresponding coefficient

of the expansion

in plane waves of the discrete

part of the spectrum

of t. In the course of the solution, the eigenvectors (2.5) will be required.

of the equation conjugate

to

If we replace d/& by -a/& in the transport equation, gin’ (E) by gp) (E’ ), by l@(E) k

@(E’)

and IJ (x, p, E) by $’ (x, FL,E), then seek rli+ (x, ,I.L,E) as

aknexpansion in T,’ &, cl, E) = ex’r at (t, IL, E), we get

km;1

n=O

Here,

z

z l-1, l] are roots of the equation det A* (z) = det p - zCt (z)] = 0,

where ii IlCf (2) llnn’llkk’=

[2(nr - i)+ IIQ~_.i(zp,+~z)

J ~~:‘-“(E)~:“-“(E)~E, n2

n’,

= \

DW- I)+ q~nq(zp,-i(z)

J~~~‘-“(E)~*‘“-*‘(E)~E, n < 12’.

As was to be expected, the discrete spectra of the direct and conjugate problems are the same. The vector Hf (z) is the solution of the equation (for admissible values of z)

(z)H’ (2) = 0

(2.9)

A*

and is defined up to a constant,

which can be put equal to unity.

The following property of the matrices X (z) and h*(z) during the course of solution:

will also be required

Some

problems

of neutron

213

transport theory

(2.10) where N and P are arbitrary

corresponding

(non-zero)

vectors,

and N and P are the rows

to them.

We thus obtain

an expansion

in a complete

set of functions

M

$f? f&E)=

(2.11)

c

b,[cI, (L,, p, E) e--XfLp +

p=*

+ iD (- L,, p, E) edL,

]+rtD(t, p,E)e-““dt, --I

where account has been taken of the symmetry of the problem, and the b,, h (t, E) and H(t) have to be determined from the boundary conditions. After substituting the following

(2.11) in (2.2) and using

the connecting

equation

(2.8)

SIE for H (t) is obtained:

where

B(t) = H(t)e”““,

IlIP

IInn’llkk’= P,+(t)*

2(n’-I)+l 3

(E) dE; 0

m

F’*‘(t)=

ff(ltl,E)h(r,E)dE--(t)Q’*‘(t)i 0

3i sB(s)

f

p

(1)

s~(eLZBir

- I)ds.

II

and the upper sign refers

to t > 0 and the lower to t < 0.

V. P. Gorelou,

214 Notice

that,

V. I. Il’in

and V. I. Yufereu

like H (t), both tit (t) and F (t) satisfy

(2.13)

the relationship

M (t) = AM (-t),

whereIIIlAll,,~ Ilkkt = t&g6,~(-lY_". As in Section Riemann contour

1, solution

boundary

of Equation

(2.12)

can be reduced

value problem on the jump of analytic

to solution

vector-function

of a

at a

(-1, 1): A+ (t)N+ (t) - h- (t)N- (t) = t8’ (t),

where h:’ (t) = hP (t) +

nitP(t>, since

CM=~~dp.

The required

and satisfies infinity,

vector

the condition

in accordance

The condition (2.10)

RC (z)l

Recalling

for being analytic

with the properties

for this last system

of the matrices

(2.14)

can be found from the system

N (z)

X (z) and X*(z),

i tF(t)

(2.13),

t-_z

outside

the cut and vanishing

of the elements

to be solvable

of equations

at

of matrix h(z).

follows

from property

and is

dt = 0.

it is easily

seen that

R+(-L~)~~dt-_R+(L,)jtFodt, P

since

A* = 1. Hence

number of equations constants

b R’

_,

half the equations remaining

is exactly

t-&l

of system

(2.14)

can be discarded;

equal to the number of unknown

the

Some problems Before

the bp, the H (t) appearing

finding

The first

approximation

N, (q

of neutron transport

=

e-O/l’l

to the vector

{w

215

the00

in F (t) has to be found.

H(t) can be found in the usual way [31:

f(ltl,E)h(t,E)dE-

[f ”

i sds -WQW]+L&)l--[f

cm

fflst,E)h(s,E)dE-

-1

-WQfs)]}

”

I

where

L,(t)

= 0.5{[A+(t)]-’

L,(t)

=

After this, the latter

(2nit)-‘{[h+(t)]-’

all the $

is a system

-

I-“}; [h-(t)]-‘}*

can be found uniquely

of inhomogeneous

The unknown function condition

+ [h-(t)

from the system

(2.14),

since

equations.

X (t, E) is then easily

obtained

from the boundary

(2.2): M 1 (t, E) = e-*/m {f(ltl,~)-~b,[~tLn,t,~)exp(--a/L,)t

p=i gLn)(E)X

+m(-L,,t,L‘)exp(a/L,)]-~~~1’,~(t)~ n=O

S&w(s) XY f--t s-t where a term appears, radiation;

this situation

For instance, whose energies

with the nuclei of isotropic become

t~(-Cf),

, t

with the unscattered

was mentioned that the question

of N and J (n) can only be decided

materials. neutrons

ds

connected

it is quite evident choice

e_o,n

in Section

really

with respect

satisfactorily

N = 0.

to the

for concrete

if a plate of fairly heavy material

i.e.

of the external

I.

are such that only the s-neutrons

or the mean energy

becomes necessary

neutrons

of convergence

of the medium, then we can confine

collisions,

lighter

k=l

is irradiated interact

ourselves

by

effectively

to the approximation

On the other hand, as the plate nuclei of the radiation

to take into account anisotropic

neutrons

increases,

scattering,

it

which then

216

V. P. Gorelov,

becomes

significant.

simplest

case

uranium

235 (or plutonium

fission dence

The choice

is that of a plate

main features

merely

of J (n) is a more complicated

matter.

material

239), when elastic

scattering

is negligible.

are determined

by the kernel

distribution

which is energy-degenerate

of the fission

scattering

and V. I. Yuferev

of fissionable

of the energy

integral,

V. 1. ll’in

neutron

spectrum

influences

(it is usual

on the bombarding

the shape

The

with high enrichment

of the spectrum

in

Then,

to neglect

the depen-

neutron).

Inelastic

at small

the

of the

energies,

when

the relative number of neutrons is small. We can therefore confine ourselves to small values of J(0) (n = N = 0, since in transport theory fission and inelastic scattering

are usually

We shall

assumed

isotropic

next turn to extending

with respect

the technique

to release

for solving

of neutrons).

the transport

equation described in [31, to the case of a multi-layer sphere, in individual layers of which linear anisotropic scattering has to be taken into account. 2. DETERMINATION

OF THE

CRITICAL

DIMENSION

OF A

MULTI-LAYER SPHERE WHEN LINEAR ANISOTROPIC SCATTERING IS TAKEN INTO ACCOUNT We shall confine

ourselves

here to the single-velocity

In [31, the method of generalized transport

equation

isotropic

collisions

The technique

in a multi-layer

eigenfunctions plate

approximation.

was used to solve

and a sphere;

the approximation

the of

was used in the sphere. of solution

in [3] was somewhat

different

to that described

above. It will be shown below that the technique of [3] can be extended to the case when anisotropy is significant up to a linear term in individual layers. Notice

also that a common feature

geometry since

is the assumption

of .the use of Case’s

of constant

only then is it possible

to reduce

method in a spherical

free path length the transport

relative

equation

to the radius, in the sphere

to

an equation formally resembling the equation in a plane geometry, to which the method of generalized eigenfunctions can be applied. The same sort of situation applies

as regards

allowing

for anisotropy

only up to a linear

term [7].

To sum up, our problem is to find the dimension of the active, e.g. central, part of a laminated sphere, in individual layers of which anisotropic scattering is significant. In the approximation general form

described,

the transport

equation

can be written

in the

V. P. Gorelov,

V. I. Il’in

217

and V. I. Yuferev

where fl is the unit vector along the direction of neutron motion, z, (r) and c,(r) are the scattering and fission cross-sections respectively (piecewiseconstant functions of the radius vector r), v is the number of neutrons involved in each act of fission,

and the other notation is as before.

The solution of (2.15) must be positive,

and must also satisfy

the conditions

~,!f(r, 12) is continuous on the interfaces, (2.16)

$(RN’

Q>= 0, Q?I < 0,

where R, is the radius vector of the free surface and n is the outward normal to the free surface. The function g(s1’ + 0, r), namely, the scattering

indicatrix,

is normalized

to unity in each layer:

and, from physical

considerations,

p, = Q’sl of the scattering

must be a positive

angle.

This function can be written explicitly, anisotropic

scattering,

function of the cosine

in the approximation of linear

as

gw

+ C!, r) = 1 + 3&r)

where F,,(r) is the mean cosine of the scattering

(G’Q>, angle (it is a piecewise

constant function of r). By using the method employed in [7] for a homogeneous sphere, Equation (2.15) under conditions (2.16) can be reduced to the integral equation

218

V. P. Gorelov,

-Riy
+1(P), Here,

(P) =

V. I. Il’in and V. 1. Yuferev

pan

=

P\ $(F, Q)&?

G(P)+-Z(P)

c(P>=~

is an odd function

of p,

;

I:

p(p)=

3PobMP) t~--c(P)l;

r:

-JL(IRi-,+ pl)] andf @I = 4-C-p); R k

(2.17)

MR==

E, (x) is the integral lengths. Using

(2.18)

@(P, u)=

i,k==iJ

s PdPWP* Rh--i exponential,

the definition

T(P)--f(P)=

of integral

)...)

N;

and all dimensions

exponential,

~~~P,~)~~, . ..-I

are expressed

it is easily

in free path

found that

219

Some problems of neutron transport theory

RN

1

--z .OJ c(p’)

B(P’)

[

1 +------ix2 c (P’)

(P- P’!

I 1 exp

-

I

2

cp(P’)dP’,

o>u>-1. It is easily

and turns

shown that @ @, u) satisfies

out to be an antisymmetric

the equation

function:

@(p, u>= -dr c-p,-u>. This

function

must vanish

+ R,,

must be continuous

on the medium interfaces,

and with p =

with u % 0 respectively.

Within one layer, where all the parameters are constant, the solution of the inhomogeneous equation (2.19) will be sought as the sum of solutions of a homogeneous equation and a particular solution of the inhomogeneous equation. As before,

the solution

superposition

of the homogeneous

equation

will be sought

as a

of plane waves

TVi 07, U) = .“”

‘i (‘, ‘),

where we use the normalization t

J hi(Y, ZZ)dZS=1.

--1

The system of functions hi (v, U) again splits into two classes, and may be shown to be complete; but it is now a system of orthogonal functions, as distinct

(2.20)

from problems

involving

energy

dependence,

in fact,

iMY,U)Jli(V’,@l udu_Ov’f Y. J_*(1+(pa/ci>uz) -’

V. P. Gorelov, V. 1. Win and V. 1. Yuferev

220

Within say the i-th layer, thus be written

the solution

of the homogeneous

equation

moi(p,u)=aji’hli’(e)orp(-p/v,)+al-)h,o(u)eap($)+

(2.21)

may

as

1

where

hj*) (u)

= -

are the ei~enfunctions

CiVj

2

(vi F U)

of the continuous

w

hj(V, u)=~-

are the eigenfunctions

(1 +(f3i/Ci)U2)

part of the ~1spectrum,

(1 +(Pi~ci)~z) +

2

v-u

of the continuous

Xi (v), in which we have to replace

Use has been made here of the results Section,

on the indicatrix

there only exist

pairs

of

TSI,where

of discrete

sought

by means

of the inhomogeneous of Green’s

while the integration (Ri_l,

Rib

of Part 2 of this The coefficients

eigenvalues.

equation

within

lnl(1

it was shown that,

posed at the start

(2.21) and the Hi (v) have to be found from the boundary The solution

u)

I+$+$];

and &vi are the zeros of the function v)/ (1 - Y)] by In t(v + I)/‘+ - I)].

under the conditions

_

1

part of the 1) spectrum;

h&+1-$q(ln~)(

(2.22)

n.(v)s(v

in

conditions. the i-th layer will be

function

over Vi splits

into integration

either

over the interval

or over (Ri_l, Ri)*

To obtain Gi @, u,/‘p,,, u,), we use the method described in [91 in reference to the transport equation in a plane geometry; the relevant expressions are derived in our Appendix. Notice that aHi @, U) is antisymmetric, i.e.,

-t

Some problems

The complete

solution

since

possesses

and, -u;(-‘,

it also

H,(V) After

of neutron

of (2.19)

within

transport

the i-th

antisymmetry,

221

theory

layer

may be written

it necessarily

follows

as

that

ai

=

= Hi (-V).

substituting

SIE can be obtained

(2.23)

each

ai @, u) in the boundary

for Hi (v), in just

E.(u)B(u)+C,(lul,u)pg-

the same

and edge

way as was

dv = F,

--I

conditions,

done

an

in [31:

(u),

where I(B(U>IIi= Hi (u) exp (Ri/, 1~1);

The matrix

form of the vector

h (u) is given

matrices

C,(lul,

matrices

in [31, except

contain path

a factor

lenths.

Q”‘(u)

in the Appendix.

quai

hi (u) is

Cz(‘) ( v, u) are the same

u) and

that

the elements

(1 +- t’Ji/ci)uz),

Notice

is given

in [31, where

to (2.22). as those

corresponding

and that

The

form of the

The elements

of the corresponding to a specific

all dimensions

of the

layer

are expressed

in free

that

F,(u) = -F,(-u). As above,

the upper

sign

in (2.23)

corresponds

to u > 0, and the lower

to

u < 0. By using above,

a technique

the following

described

condition

in [31, and somewhat

for solvability

different

of the initial

to that

SIE (2.23)

used

can be

obtained: N

eip(IvI)FOp(v) vki i c --i p=i

(2.24) where such

eip (1~1) is a combination that

~0,

(vi*-~~)-~d~

of exponents,

det \/eip (IV\) (1 f 0 in the same

bounded

interval.

k, = 1,2, in the interval

(-1, 1) and

V. P. Gorelov,

222

V. 1. Il’in

and V. I. Yuferev

of eip (IV\) can be found in concrete

The form of the elements the relationship [31

problems

from

where

i-1

is

a bounded

function.

It can easily be seen, on recalling that FOp (v) = \(F,(v)(\,, is odd, that Equation (2.24) is satisfied as an identity for even hi = 2. Hence we finally have in this equation ki = 1. In this case the number unknown constants oi(+‘. Before finding zero approximation

of equations

(2.24) is the same as the number of

we have to find S (v), as was shown in [31. The the ai( 3 (v) = 0 can sometimes prove useful.

After finding B (v), the system of equations (2.24) gives us all the ai in Finally, by using (2.18) and (2.17), a terms of linear combinations of M,. system of homogeneous algebraic equations is obtained in the M,. The condition for this system to be solvable enables the critical dimension of the system to be found,

and hence,

all the quantities

Such are the special velocity

approximation

equation

involving

features

defining

of this problem,

and the somewhat

the jump of an analytic

different

4,(p). resulting

from the single-

technique

for solving

the

function.

APPENDIX

‘;i+‘(uo)hl+)(u)exp (-5)

+

1

+f

0

Gi(~9u/x0,

~0)

=

(

gt(Uo,V)hi(V, U)eXp

(-5)

dv,

x >

~0;

Some

problems

of neutron

(-) -gi

C-J

(

223

theory

x -

(u)exp

(uO)hi

I-f

transport

x0

)

V*

-

0

i?i(uo,V)hi(V,U)eXp

(-5)

dv,

x <

x0;

--i

ii*)

ht*‘( uo)

= 1 2N*(*’

(1 +

(pi/Ci)

1 gi

(UO,

V)

=

hi(V,

u0)

~ 2Ni Cz2Vi2

$*)

UO’)

(v)

(1

2Vi

+

(fii/Ci)

(2

(j3i/Ci)

UO’)

-

=*-

2

N,(v) =

(2

1-

1)

+

vi2

(3vt($)

+(p~,ci)v?) {hi2(v)+[ yy

where (8)

@Oj (Z, U) =

’

(+I Ui

c

h,(+)(u)exp

+I)

If

ln$f+],

($) vq} !

(--f1

(ii)I.

-hhl-'(u)exp

Vi

Vi

Translated

by D. E. Brown

REFERENCES

1.

CASE, K. M. Elementary solutions Ann. Phys., 9, l-23, 1960.

of the transport

2.

BEDNARZ, R. J. and MIKA, J..R. Energydependent geometry, J. Math. Phys. 4, 1285-1929, 1963.

3.

GORELOV, V. P. and YUFEREV, V. I. Solution of the single-velocity equation of neutron transport in multi-layer plane and spherically symmetric systems, Zh. uychisl. Mat. mat. Fiz., 11, 1, 129-136, 1971.

4.

MIKHLIN, S. G. Lectures on Linear Integral Equations (Lektsii integral’nym uravneniyam), Fizmatgiz, Moscow. 1959.

5.

WHITTAKER. E. T. and WATSON, u. P.. 1940.

G. N.

A Course

equation

and their applications,

Boltzmann

in Modem

equation

in plane

po lineinym

Analysis,

Cambridge

224

V. P. Gorelov,

V. I. Il’in and V. 1. Yuferev

6.

GAKHOV, 1963.

7.

MARCHUK, G. I. Methods of Designing Nuclear Reactors yadernykh reaktorov), Atomizdat, Moscow, 1961.

8.

HAUNGS, Von G. Das Spektrnm der monoenergetischen leichung in ebener Geometrie mit linear-anisotroper energie, 11, 1924, 1966.

9.

MIKA, J. R. 415-427,

F. D.

Boundary

Value Problems

Neutron transport 1961.

(Kraevye

with anisotropic

zadachi),

Fizmatgiz,

Moscow,

(Metody rascheta

stationaren BoltzmanngStreufunktion, Atomkem-

scattering,

Nucl.

Sci. Engng.,

11,